Use a graphing utility to graph the function on the closed interval [a, b]. f(x)
ID: 2895082 • Letter: U
Question
Use a graphing utility to graph the function on the closed interval [a, b]. f(x) = x 2 sin x 6 , [1, 0] Determine whether Rolle's Theorem can be applied to f on the interval. (Select all that apply.) Yes, Rolle's Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because f is not differentiable in the open interval (a, b). No, because f(a) f(b). If Rolle's Theorem can be applied, find all values of c in the open interval (a, b) such that f'(c) = 0. (Enter your answers as a comma-separated list. Round your answers to four decimal places. If Rolle's Theorem cannot be applied, enter NA.)
Explanation / Answer
f ' (c) = [ f(0) - f(-1) ] / [ -1 - 0 ]
(1/2) - cos( * c/6) * (/6) = [ 0/2 - sin( * 0/6) - ( -1/2 - sin( * -1/6) ) ] / [ -1 ]
(1/2) - (/6)cos(c/6) = - [ 0 - ( -1/2 - sin(-/6) ) ]
(1/2) - (/6)cos(c/6) = - [ 1/2 + sin(-/6) ) ]
(1/2) - (/6)cos(c/6) = - [ 1/2 - sin(/6) ) ]
(1/2) - (/6)cos(c/6) = - [ 1/2 - (1/2) ]
(1/2) - (/6)cos(c/6) = - [ 0 ]
(1/2) - (/6)cos(c/6) = 0 <---- yes it can be applied.
(1/2) = (/6)cos(c/6)
6/(2) = cos(c/6)
3/ = cos(c/6)
cos^-1( 3/ ) ; 2n - cos^-1( 3/ ) = c/6
6cos^-1( 3/ )/ ; 12n - 6cos^-1( 3/ )/ = c/6
- 6cos^-1( 3/ )/ -0.5759
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